have tended to emphasize only one aspect of proficiency, with the expectation that other aspects will develop as a consequence. For example, some people who have emphasized the need for students to master computations have assumed that understanding would follow. Others, focusing on students’ understanding of concepts, have assumed that skill would develop naturally. By using these five strands, we have attempted to give a more rounded portrayal of successful mathematics learning.
The overriding premise of this book is that all students can and should achieve mathematical proficiency. Just as all students can become proficient readers, all can become proficient in school mathematics. Mathematical proficiency is not something students accomplish only when they reach eighth or twelfth grade; they can be proficient regardless of their grade. Moreover, mathematical proficiency can no longer be restricted to a select few. All young Americans must learn to think mathematically if the United States is to foster the educated workforce and citizenry tomorrow’s world will demand.
(1) Understanding: Comprehending mathematical concepts, operations, and relations—knowing what mathematical symbols, diagrams, procedures mean.
Understanding refers to a student’s grasp of fundamental mathematical ideas. Students with understanding know more than isolated facts and procedures. They know why a mathematical idea is important and the contexts in which it is useful. Furthermore, they are aware of many connections between mathematical ideas. In fact, the degree of students’ understanding is related to the richness and extent of the connections they have made.
For example, students who understand division of fractions not only can compute . They also can represent the operation by a diagram and make up a problem to go with the computation. (If a recipe calls for cup of sugar and 6 cups of sugar are available, how many batches of the recipe can be made with the available sugar?)
Students who learn with understanding have less to learn because they see common patterns in superficially different situations. If they understand the general principle that the order in which two numbers are multiplied doesn’t matter—3×5 is the same as 5×3, for example—they have about half as many “number facts” to learn. Or if students understand the general principle that multiplying the dimensions of a three-dimensional object by a factor n increases its